Let C and D be digraphs. A mapping ?: V(D)→ V(C) is a C-colouring if for every arc uv of D, either f(u)f(v) is an arc of C or f(u) = f(v), and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely C-colourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for every digraph D that is uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number r ≥ 1, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.
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